Well-posedness and longtime behavior of the phase-field with memory in a history space setting

A thermodynamically consistent phase-field model with memory, based on the linearized version of the Gurtin-Pipkin heat conduction law, is considered. The formulation of an initial and boundary value problem for the phase-field evolution system is framed in a history space setting. Namely, the summed past history of the temperature is regarded itself as a variable along with thetemperature and the phase-field. Well posedness results are discussed, as well as longtime behavior of solutions. Under suitable conditions, the existence of an absorbing set can be achieved.